Vector Maps as Positive Evidence in Crystal Analysis

Abstract

THE principle of crystal analysis by means of vector maps may be indicated briefly as follows. If the electron density throughout a crystal is expressed by a triple Fourier series ρ whose typical coefficient is the complex number a, the triple Fourier series V whose typical coefficient is the real number |a|2 has two properties: (1) the coefficients in V are determinable from and collectively resume all the information given by observations of intensities of reflected X-rays ; (2) a maximum value of V corresponds in position to a step between positions for which ρ is a maximum. Since the points at which maximum values of ρ occur are the individual atoms, we can say that the observed data are made to furnish directly the vector map of the atomic structure. The ultimate usefulness of the method depends, therefore, theoretically on the extent to which a vector map S2 determines, or at least characterizes, the nuclear structure from which it is derived, practically on the efficiency of a mathematical technique for achieving such deduction as is theoretically within reach. Until recently, it was thought that the direct value of the vector map was small and that the map was in effect only a test to be applied to structures suggested independently, and by showing that the main features of the atomic structure can be read from the map and that in many cases every detail of the atomic structure is implicit in the details of the map, Dr. Wrinch has opened a new chapter in crystal analysis1.